Basic equations for constructing geopotential mode

Authors: M S Petrovskaya A N Vershkov

Publish Date: 2011/12/31

Volume: 86, Issue: 7, Pages: 521-530

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Abstract

This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov J Geod 843165–178 2010 where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame The problem of converting the potential derivatives of the first and second orders into geopotential models is studied Two kinds of basic equations for solving this problem are derived The equations of the first kind represent new nonsingular nonorthogonal series for the geopotential derivatives which are constructed by means of transforming the intermediate expressions for these derivatives from the abovementioned paper In contrast to the spherical harmonic expansions these alternative series directly depend on the geopotential coefficients barC nm and barS nm Each term of the series for the firstorder derivatives is represented by a sum of these coefficients which are multiplied by linear combinations of at most two spherical harmonics For the secondorder derivatives the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics As compared to existing nonsingular expressions for the geopotential derivatives the new expressions have a more simple structure They depend only on the conventional spherical harmonics and do not depend on the first and secondorder derivatives of the associated Legendre functions The basic equations of the second kind are inferred from the linear equations constructed in the cited paper which express the coefficients of the spherical harmonic series for the first and secondorder derivatives in terms of the geopotential coefficients These equations are converted into recurrent relations from which the coefficients barC nm and barS nm are determined on the basis of the spherical harmonic coefficients of each derivative The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the leastsquares approach The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis In particular they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP GRACE and GOCE missions and the gradiometry data from the GOCE mission

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